let V be Z_Module; :: thesis: for a, b being Element of INT.Ring
for v being Vector of V holds (a - b) * v = (a * v) - (b * v)
let a, b be Element of INT.Ring; :: thesis: for v being Vector of V holds (a - b) * v = (a * v) - (b * v)
let v be Vector of V; :: thesis: (a - b) * v = (a * v) - (b * v)
thus (a - b) * v = (a * v) + ((- b) * v) by VECTSP_1:def 15
.= (a * v) + (b * (- v)) by Th5
.= (a * v) - (b * v) by Th6 ; :: thesis: verum